Consider the diffusion equation with convection on the real line. მu. at = K J²u მ2 მu. + C- მე u(0,x) = f(x) (∞0>x>∞0-) a) Take the Fourier Transform of the PDE and initial data with respect tox and solve for u(t, w). b) Using the fact that the inverse Fourier Transform of a Gaussian is F(w) = e-aw² f(x) = -x²/(4α) (a > 0) write the solution u(t, x) to the convection equation as a convolution of the initial data f(x) with an appropriate function. (HINT: You should get a term like eicut in u. This should simply shift x to x + ct in the final answer!)

point If 1 g(u) V5и + 8 then g' (u) =

dt A model for the velocity v at time t of a certain object falling under the influence of gravity in a viscous medium is given by the equation = 1- . From the direction field shown in the figure to the right, sketch the solutions with 14 the initial conditions v(0)= 9, 14, and 18. Why is the value v=14 called the "terminal velocity"? Choose the correct sketch of the solutions with the initial conditions v(0) = 9, 14, and 18. O A. O B. as t→ + ∞o. Q C Why is the value v=14 called the "terminal velocity"? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. All solutions have limiting value O B. All solutions have limiting value as t→14. OC. At the end of its fall, the object always has velocity v = 14. O C. 144

A comet passing by a star, with initial velocity (30, 60), experiences a force which causes it to accelerate according to a(t) = (-25e – 25e-t, 25e – 25e-t) (in m/sec?). Find a formula for its velocity at time t.

Find the derivative of f(w)= sin(4w²)+cos(4w²). A) Swcos(4w°)–8wsin(4w²) B) -8wcos(4w?)+8wsin(4w²) C) 4cos(4w?)–4sin(4w?) D) 4cos(4w?)+4sin(4w²)

a) What is the linearization tof(x) = ³√x at x = 125 ?

Find the velocity and acceleration vectors and the equation of the tangent line for the curve r(t) = 3 cos(t)i + sin(6t)j at t 0. (Use symbolic notation and fractions where needed. Give your answer in the form (*,*).) v(0) 1(t) = Incorrect (0,6) (Use symbolic notation and fractions where needed. Give your answer in the form (*,*). Use t for the parameter that takes all real values.) = a(0) = (3,0) +t(0,6) (-3,0)

Consider a particle moving along a curve so that it is at position f(t) = (2+³, 2t²) at time t. Then the particle is moving parallel to (1, -2) at time t at the point when the direction of motion is =

Use reiation Ship: the fcx) (x) To Calculate Js?

Suppose an object is moving with position function given by p(t) -t^3-6t^2-36t. 1. Find the velocity function at v(t) of the object 2. Find all times for which the object is at rest (i.e. has zer velocity) 3. Find the acceleration function at a(t) of the object 4. For which times is the object sppeding up?